AP Calculus AB/BC Unit 7 ReviewDifferential Equations

AP Calculus Unit 7 covers differential equations, which are equations that relate a function to its own derivative, like dy/dx = ky. The single biggest idea is that you can describe how something changes (a rate) without knowing the function itself, then recover the function using integration. The main technique is separation of variables, and the main applications are exponential growth and decay (plus logistic growth on the BC exam). Unit 7 makes up 6-12% of the AP exam for both AB and BC.

What this unit covers

Reading and writing differential equations

• A differential equation links a function and its derivative. "The rate of change of y is proportional to y" translates directly to dy/dt = ky. Turning sentences into equations like this is a tested skill on its own.
• You also go the other direction. Given a proposed function, you verify it solves a differential equation by taking its derivative and plugging both the function and the derivative back into the original equation. No solving required, just substitution and checking.
• One differential equation usually has infinitely many general solutions (a whole family of curves that differ by a constant). That "+C" from integration is what creates the family.

Slope fields, the picture of a differential equation

• A slope field draws a tiny line segment at each point (x, y) whose slope equals dy/dx at that point. It is a visual map of every possible solution curve at once.
• To sketch one, plug each point into the differential equation and draw a segment with that slope. If dy/dx = x + y, the segment at (1, 1) has slope 2.
• To read one, start at an initial condition and "follow the flow." The solution curve threads through the field, staying tangent to the segments. You can spot where solutions increase, decrease, level off, or approach a horizontal asymptote without ever solving anything.
• A classic multiple-choice move is matching a slope field to its equation. Check whether slopes depend only on x (segments identical in vertical columns), only on y (identical in horizontal rows), or both.

Euler's method (BC only)

• Euler's method approximates a solution numerically by taking small tangent-line steps. From a point, compute the slope from the differential equation, step forward by a fixed step size, and repeat.
• The update rule is new y = old y + (step size)(slope at the old point). It is just repeated tangent-line approximation, the same idea as linearization from Unit 4, applied over and over.
• Smaller step sizes give better approximations because each tangent line stays closer to the true curve.

Separation of variables and particular solutions

• Separable equations can be rewritten so all the y's (with dy) sit on one side and all the x's (with dx) sit on the other. Then you integrate both sides. For dy/dx = xy, you get dy/y = x dx, integrate, and solve for y.
• Put the +C on one side immediately after integrating, before doing any algebra. A constant added too late, or in the wrong place, is a classic point-loser.
• A general solution is the whole family of curves. An initial condition, a single known point like y(0) = 3, pins down one particular solution by determining C.
• Particular solutions can have domain restrictions. The solution must be defined on an interval containing the initial condition, so part of an algebraically valid answer may need to be thrown out.

Exponential and logistic models

• The statement "rate of change is proportional to the quantity" gives dy/dt = ky, and its solution is y = y₀e^(kt). Positive k means growth, negative k means decay. This one model handles populations, radioactive decay, and continuously compounded interest.
• You should be able to derive y = y₀e^(kt) by separating variables, not just memorize it. The derivation is a fair FRQ ask.
• Logistic growth (BC only) modifies the exponential model so growth slows near a cap. The equation dy/dt = ky(a − y) says the rate is jointly proportional to the quantity and to how far it is from the carrying capacity a.
• For logistic problems, you usually do NOT solve the equation. You interpret it. The carrying capacity is the limiting value as t goes to infinity, and the population grows fastest when y is exactly half the carrying capacity.

Unit 7, Differential Equations at a glance

Why Unit 7, Differential Equations matters in AP Calc

Unit 7 is where differentiation and integration finally work together on the same problem. You are handed information about a rate and asked to rebuild the function, which is the whole point of the Fundamental Theorem played out in modeling form. It is also the most "real-world" unit in the course, because nearly every natural process (population, decay, cooling, motion) gets described first as a rate.

• It cements the analysis-of-functions big idea. A differential equation IS a statement about a derivative, so everything you know about increasing, decreasing, and concavity applies to its solutions.
• It is the payoff for integration skills. Separation of variables only works if you can actually integrate both sides, including u-substitution.
• It builds the modeling habit the exam rewards, going from a verbal description to an equation to a solution to an interpretation in context with correct units.

How this unit connects across the course

• Implicit differentiation (Unit 3) is the engine behind verifying solutions and behind separation of variables itself, since you treat y as a function of x throughout.
• Tangent-line approximation (Unit 4) is exactly what Euler's method repeats step after step, and related-rates thinking is the same "translate a sentence about rates into an equation" skill.
• Antidifferentiation and u-substitution (Unit 6) do the actual computation once variables are separated. The accumulation form y₀ + ∫ from a to x of f(t) dt is itself a particular solution to dy/dx = f(x).
• BC students will see the logistic equation again in Unit 10, and slope-field-style reasoning about rates returns with parametric and vector-valued motion in Unit 9.

Key formulas and procedures

• dy/dt = ky models exponential growth or decay whenever a rate is proportional to the quantity itself.
• y = y₀e^(kt) is the solution to dy/dt = ky with y = y₀ at t = 0. Be able to derive it by separating variables.
• Separation of variables: rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, integrate both sides, add C right away, then solve for y and apply the initial condition.
• Verifying a solution: differentiate the candidate function, substitute y and y′ into the differential equation, and confirm both sides match.
• Slope field sketching: at each given point, evaluate dy/dx and draw a short segment with that slope.
• Euler's method (BC): y_(n+1) = y_n + Δx · f(x_n, y_n). Each step rides the tangent line for a distance Δx.
• F(x) = y₀ + ∫ from a to x of f(t) dt is the particular solution to dy/dx = f(x) satisfying F(a) = y₀. Useful when the antiderivative has no nice closed form.
• Logistic model (BC): dy/dt = ky(a − y), where a is the carrying capacity. The limit of y as t → ∞ is a, and dy/dt is largest when y = a/2.

Unit 7, Differential Equations on the AP exam

Unit 7 is 6-12% of the exam for both AB and BC. In multiple choice, expect slope-field matching (pair an equation with its field, or a field with its solution curve), quick separation-of-variables solves, verification questions where you test whether a given function satisfies an equation, and Euler's method computations on the BC exam.

In free response, differential equations show up regularly, often as a multi-part question that combines several skills. A typical structure asks you to sketch solution curves on a given slope field, write a tangent line approximation or run Euler's method (BC), reason about concavity by finding d²y/dx² from the differential equation, and finish by finding the particular solution with separation of variables. Logistic prompts on the BC exam usually test interpretation, asking for the carrying capacity or the value of y where growth is fastest, without requiring a full solve. Show every step of separation clearly. Readers award points for separating, antidifferentiating each side, the constant of integration, using the initial condition, and the final solved form, so skipped steps cost real points even when the answer is right.

Essential questions

• How can you describe a quantity's behavior when all you know is its rate of change?
• Why does one differential equation have infinitely many solutions, and what makes a particular solution unique?
• What does it mean, mathematically, for growth to be "proportional to the amount present," and why does that always produce an exponential?
• How can you predict a solution's long-term behavior (like a carrying capacity) without ever solving the equation?

Key terms to know

• Differential equation: an equation relating a function of an independent variable to one or more of its derivatives.
• General solution: the family of all functions satisfying a differential equation, including an arbitrary constant C.
• Particular solution: the single solution from the family that passes through a given initial condition.
• Initial condition: a known point, such as y(0) = 5, used to determine the constant in a general solution.
• Slope field: a grid of short line segments showing the slope dy/dx at sample points in the plane.
• Separation of variables: the technique of moving all y terms with dy to one side and all x terms with dx to the other, then integrating.
• Euler's method: a BC-only numerical procedure that approximates a solution curve with repeated small tangent-line steps.
• Step size: the fixed horizontal distance Δx of each Euler's method step; smaller steps mean better accuracy.
• Exponential growth and decay: the model dy/dt = ky with solution y = y₀e^(kt); k > 0 grows, k < 0 decays.
• Logistic growth: the BC-only model dy/dt = ky(a − y), where growth slows as y approaches the carrying capacity.
• Carrying capacity: the limiting value a logistic solution approaches as t goes to infinity.
• Domain restriction: the requirement that a particular solution be defined on an interval containing the initial condition, which may exclude part of an algebraic answer.

Common mix-ups

• General vs. particular solutions: y = Ce^(kt) is a family of curves; y = 5e^(2t) is one curve. If the problem gives an initial condition, the answer must have no C left in it.
• Adding C too late: the constant appears the moment you integrate, before any exponentiating or algebra. Writing y = e^(kt) + C instead of y = Ce^(kt) is wrong, and it comes from adding C after solving for y instead of before.
• Exponential vs. logistic: dy/dt = ky grows without bound, while dy/dt = ky(a − y) levels off at a. If a problem mentions a cap, limit, or carrying capacity, it is logistic.
• Euler's method vs. one tangent line: a single tangent-line approximation uses one slope for the whole jump. Euler's method recomputes the slope at every step, which is why it is more accurate over the same interval.